Delta Kernel Ledger Em Right

The ledger union rule

A ledger, in this framework, is a discrete record of which classical logical principles a derivation consumed. The record has four Boolean fields: full excluded middle (EM), the limited principle of omniscience (LPO), Markov's principle (MP), and a separate induction tier flag. The declaration em_right is a theorem about combining two such ledgers with a union operation. It states that if the second ledger has the EM field set to true, then the union of the two ledgers also has EM set to true. In plain terms: once a derivation has used full excluded middle, merging it with any other derivation's assumptions cannot erase that fact.

The union operation itself is a pointwise logical OR across the four fields. The lemma is proved by unfolding the definitions and checking the Boolean cases; it is a structural fact about the ledger as a data type, not a claim about mathematics or physics. Its role is bookkeeping: it guarantees that when two derivations are combined, the resulting ledger correctly reports every posit either one consumed. The companion lemmas em_left, lpo_left, lpo_right, mp_left, and mp_right do the same for the other fields and sides.

What the declaration does not claim is more interesting than what it proves. It does not say that using EM is ever forced, nor that a derivation carrying EM is invalid. The ledger distinguishes forced judgments, which consume no posits, from conditional ones, which do. A theorem like ofEM_isForced shows that a ledger with EM set is not forced, but em_right itself is silent on that distinction. It merely records that EM, once used, propagates through union. The framework's own stratification treats whether full induction is forced as a measured question, not a pre-judged one; this lemma is a small piece of that accounting machinery, not a verdict on the principle it tracks.

theorem em_right {a b : Ledger} (h : b.em = true) : (a.union b).em = true := by
  simp [union, h]

/-- Merge two ledgers (a derivation consumes the posits of all its parts). -/
def union (a b : Ledger) : Ledger :=
  ⟨a.em || b.em, a.lpo || b.lpo, a.mp || b.mp, a.indFull || b.indFull⟩

@[simp] theorem ofEM_isForced : ofEM.isForced = false := rfl

What this page does not claim

The declaration does not prove that excluded middle is ever necessary for a derivation. The declaration does not say that a derivation using excluded middle is invalid or weaker than one that does not. The declaration does not address whether full induction is forced; that question is measured, not settled here.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/DeltaKernel/Ledger.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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